Gapless Topological Peierls-like instabilities in more than one dimension

A periodic lattice distortion that reduces the translational symmetry folds electron bands into a reduced Brillouin zone, leading to band mixing and a tendency to gap formation, as in the Peierls transition in one-dimensional systems. However, in higher dimensions, the resulting phase can present topological obstructions preventing a complete gap opening. We discuss two different mechanisms for such obstructions, emergent Weyl nodes and symmetry protected band crossings. Based on density-functional calculations, we show these mechanisms are at play in trigonal PtBi${}_2$.